Relationships between WORK, HEAT, and ENERGY
Consider a force, F, acting on a block sliding on a
frictionless surface
x
x1
M
x2
F
Frictionless
surface


dv
M
F
dt
dv
M
 Fx = F
; v  mass velocity in x  direction
dt
dv
dv
 dv dx 
FM
 M    M v
 dx dt 
dt
dx
Fdx  Mvdv
Integrating both sides from block position 1 to 2
2
2
1
1
 Fdx  M  vdv
Mv22 Mv12

 Fdx 
2
2
1
2
 W1 2  KE
∫Fdx is the energy transferred to the block in the process
 work done W1-2 by force F
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Consider an object falling in a gravitational field
m
y
F=mg
h1
h2
2
 
  F  dy    (mg )dy  mg (h1  h2 )
2
W1 2
1
1
Gravity has the potential to do work and the quantity
mgh is therefore called the potential energy
Work done by gravity results in a drop in potential energy
of the object
since W1 2  KE (see previous example)
mv22 mv12
mg (h1  h2 ) 

2
2

PE  KE
Mass PE is converted to KE via the work done by gravity
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Energy Transfer by Work
In general, work done is evaluated by
 
W1 2   F  ds
2
1
Work is a means of transferring energy, it does not refer
to what is being transferred or stored within the system.
The value of W1 2 depends on the details of the
interaction taking place between the system and the
surroundings during a process, e.g., F(s), and not just the
initial and final state
By definition a state property is evaluated at a specific
time and is independent of the process
 energy is a property of the system
 work is not a property of the system
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The differential of a property is “exact” since it is
independent of details of the process, e.g.,
2
 dE  E 2  E1
1
Differential of work is “inexact,” the following integral
can’t be evaluated without knowing details of the process
2
 W  W
2
not
 W  W2  W1
1
1
The work done over a period of time is:
2 
   2  ds 

W   F  ds    F  dt   F  v dt
dt 
1
1
1
2

where v is velocity
The rate of energy transfer by work is called power and
is denoted by W . In general,
 

W  F v
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Expansion and Compression Work
Consider the expansion of the gas in a piston-cylinder
assembly ( Pp is average pressure on piston face)
Pp
Pp
x
x1
x2
Piston Area A
2
  2
  F  ds   ( Pp A)dx   Pp dV
2
W1 2
1
1
1
For a slow or quasi-equilibrium process all the states
through which the system passes are considered
equilibrium states and thus the intensive properties, i.e.,
pressure, are uniform throughout the system PP  Pgas , so
V2
W1 2   Pgas dV
V1
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Graphical Interpretation:
State 1
P1
δW=PdV
Pressure
Process
path
State 2
P2
V1
V2
dV
x1
W  PdV
2
x2
shaded area
V2
W1 2   W   PdV
1
Volume
total area under curve
V1
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Consider two processes with the same start and end state
P1
State 1
Path 1
Path 2
State 2
P2
V1
V2
Since the area under each curve is different the amount of
work done for each path is different.
V2
V2
V1
V1
(  PdV ) path 1  (  PdV ) path 2
Work done depends on the path taken and not just the
value of the end states.
Work is not a property!
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Polytropic Compression and Expansion
The pressure-volume relationship can be described by
PVn= constant c
n= constant
The work done is:
V2
W1 2
V2
V2
c
  PdV   ( n )dV   (cV  n )dV
V1
V1 V
V1
 V2  V1 
V 1  n  V2
 c

c


 V1


1

n
1

n


1 n
1 n
n
n
but c=PV
 PV
1 1
2 2
W1 2
n
1 n
1 n
1 n
PV

PV
V




V

V


2
2
2
2
1
n
2
1
 PV

 
2 2 
 1 n  
1 n

 PV  PV
1 1 
W1 2   2 2



1 n
n 1
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For n=1
V2
 c
  PdV     dV clnV 
V1
V1
V1  V 
V2
W1 2
P = c/V
V2
 V2 
 clnV2  lnV1   c ln 
 V1 
V 
W1 2  P1V1 ln 2 
 V1 
n=1
Special case:
For n = 0
V2
P = c  constant pressure process
W1 2   PdV  P(V2  V1 )
n0
V1
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Spring Potential Energy
F
=
x
F – spring force = kx
k – spring constant (N/m)
x – displacement from
relaxed position
  2
 kx 2  x2
W1 2   F  ds   ( kx )dx  

2
1
1

 x1
1
 k ( x22  x12 )
2
2
Spring PE 
1 2
kx
2
The spring potential energy can be grouped in with
gravitational potential energy.
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Other forms of Energy
In engineering, the change in total energy of a system is
considered to be made up of macroscopic contributions
such as changes in KE and gravitational PE of the system
as a whole relative to an external coordinate frame and
Internal Energy, U.
E2- E1= (KE2- KE1) + (PE2- PE1) + (U2- U1)
Consider the vigorous stirring of a fluid in a well
insulated tank
Well insulated
Fluid
Electric
motor
system
W
Energy is transferred into the system via work by the
paddle wheel, results in an increase in the system energy.
E2- E1= (KE2- KE1) + (PE2- PE1) + (U2- U1)= W
This transferred energy does not increase the KE or PE of
the system.
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The change in system energy can be accounted for in
terms of internal energy of the fluid.
Changes in internal energy for solids, liquids, and gases
are evaluated using empirical data, e.g. U = f(T)
Microscopic Interpretation of Internal Energy
Energy is attributed to the motions and configuration of
the individual molecules, atoms and subatomic particles
making up the matter in the system.
Energy on molecular level associated with:
- Translation
- Rotation
- Vibration
- Molecular bonds
Energy on atomic level:
- Electron orbital states
- Nuclear spin
- Nuclear binding
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Conservation of Energy for Closed System
A closed system can interact with its surroundings via
work as well as thermally
Energy can be transferred between the system and the
surroundings by thermal (heat) interactions
A process that involves work interactions but does not
involve thermal interactions is called an adiabatic
process
A process that involves thermal interactions is called a
nonadiabatic process
It has been shown experimentally that the net work done
by, or on, a closed system undergoing an adiabatic
process depends solely on the end states and not on the
details of the process.
E2 – E1 = -Wad
Sign convention for energy transfer by work:
Work done by the system is positive
Work done on the system is negative
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For a quasi-equilibrium adiabatic gas compression or
expansion process the value of the polytropic exponent n
is fixed (n =1.4 for air) and thus the area under the curve
(work done) depends only on the end states
P1
1
Adiabatic path
PV1.4 = const (air)
2
P2
V1
V2
Consider an adiabatic process and nonadiabatic process
between the same two end states 1 and 2
P1
1
Adiabatic (only work)
PV1.4 = const.
Nonadiabatic (work and heat)
PVn = const.
2
P2
V1
V2
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Since the area under the two curves is different the work
done for each path is different, so Wad  Wnonad
Since the end states for both processes are the same the
system would experience exactly the same energy change
in each of the processes, so
(E2 – E1)ad = (E2 – E1)nonad = E2 – E1
We know the energy change for the adiabatic process is
E2 – E1 = -Wad
But since Wad  Wnonad we can infer that
E2 – E1  -Wnonad
Since energy must be conserved the net energy transferred
to the system in both processes must be the same. It
follows that the heat interaction in the nonadiabatic
process must involve energy transfer. The amount of
energy transferred to the closed system by heat is Q
E2 – E1 = -Wnonad + Q
The First Law of Thermodynamics states:
E2 – E1 = Q - W
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Energy Transfer by Heat
The quantity Q in the First Law accounts for any energy
transferred to a closed system during a process by means
other than by work.
Such energy transfer Q is induced only as a result of a
temperature difference between the system and the
surroundings and occurring in the direction of decreasing
temperature, e.g. heat transfer: conduction, convection,
radiation
Sign convention for energy transfer by heat:
Heat transfer to the system is positive
Heat transfer from the system is negative
Consider the immersion of a lump of hot metal initially at
Tm into a colder fluid at Tf
Tm
Tf
Tm > Tf
Tf
Q
TmTf
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Because the metal is at a higher temperature than the fluid
energy is transferred from the metal to the fluid, Q is
negative.
Since there is no work done and the change in KE and PE
is negligible, the amount of heat transferred from the
metal to the fluid is equal to the decrease in the metal
internal energy,
U  KE  PE  Q  W
Q = U1 – U 2
U2 - U1 = (-Q) or
Just like work, heat is not a property and the amount of
energy transfer depends on the process details, therefore
2
Q1 2   Q
1
The rate of heat transfer is denoted by Q and the total
energy transferred via heat over a period of time is
2

Q1 2   Qdt
1
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